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5
Condition for Gradient Filter
The condition for determining the amplitude of the gradient filters is obtained next.
The following expression represents the gradient of function
k
(
x
,
y
)
in direction
of
θ
.
cos
k
∂
∂
∂
∂
θ
x
+
sin
θ
(
x
,
y
)
(38)
y
In the Fourier domain, this equation is transformed to
2
π
i
(
u
cos
θ
+
v
sin
θ
)
K
(
u
,
v
)
.
(39)
are gradient filters in the 0
◦
,
60
◦
and 120
◦
direc-
As
h
a
(
x
,
y
)
,
h
b
(
x
,
y
)
and
h
c
(
x
,
y
)
tions, the following conditions hold:
H
a
(
u
,
v
)=
2
π
iu
(40)
i
1
v
√
3
2
H
b
(
u
,
v
)=
2
π
2
u
+
(41)
i
v
√
3
2
1
2
u
H
c
(
u
,
v
)=
2
π
−
+
.
(42)
It thus follows that
m
,
n
H
mn
(
u
,
v
)=
2
π
iu
(43)
a
i
1
v
√
3
2
m
,
n
H
mn
(
u
,
v
)=
2
π
2
u
+
(44)
b
i
v
√
3
2
1
2
u
m
,
n
H
mn
(
u
,
v
)=
2
π
−
+
,
(45)
c
must hold for any
u
and
v
. By taking the limit as
0 and using first-order
approximations for the trigonometric functions, makes it possible to rewrite these
expressions as
|
u
|, |
v
|→
m
,
n
a
mn
π
mu
=
2π
iu
4
i
(46)
√
3
2
√
3
2
1
2
u
1
2
u
m
,
n
a
mn
π
m
(
4
i
+
v
)=
2
π
i
(
+
v
)
(47)
√
3
2
√
3
2
1
2
u
1
2
u
m
,
n
a
mn
π
m
(
−
4
i
+
v
)=
2
π
i
(
−
+
v
)
.
(48)
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